Trigonometry Functions Formulas and Notes

Trigonometric Functions

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x	-90°	-60°	-45°	-30°	0°	30°	45°	60°	90°
sin(x)	\displaystyle - 1	\displaystyle - \frac{ \sqrt{3} }{2}	\displaystyle - \frac{ \sqrt{2} }{2}	\displaystyle - \frac{ \sqrt{1} }{2}	\displaystyle 0	\displaystyle \frac{1}{2}	\displaystyle \frac{ \sqrt{2} }{2}	\displaystyle \frac{ \sqrt{3} }{2}	\displaystyle 1
cos(x)	\displaystyle 0	\displaystyle \frac{ \sqrt{1} }{2}	\displaystyle \frac{ \sqrt{2} }{2}	\displaystyle \frac{ \sqrt{3} }{2}	\displaystyle 1	\displaystyle \frac{ \sqrt{3} }{2}	\displaystyle \frac{ \sqrt{2} }{2}	\displaystyle \frac{ \sqrt{1} }{2}	\displaystyle 0
tan(x)	-	\displaystyle - \sqrt{3}	\displaystyle - 1	\displaystyle - \frac{ \sqrt{3} }{3}	\displaystyle 0	\displaystyle \frac{ \sqrt{3} }{3}	\displaystyle 1	\displaystyle \sqrt{3}	-

\displaystyle f(x)= \sin(x), \displaystyle g(x)= \cos(x), \displaystyle h(x)= \tan(x)

Vertical Shift
These are functions in the form:
\displaystyle f(x)= \sin(x)+p, f(x)= \cos(x)+p, f(x)= \tan(x)+p
When \displaystyle p > 0, the function shifts vertically upwards by p units.
When \displaystyle p < 0, the function shifts vertically downwards by p units.
Example
\displaystyle f(x)= \sin(x)+3, g(x)= \cos(x)+3, h(x)= \tan(x)+3.
All the function shift vertically upwards by 3 units. Below are the graphs.

\displaystyle f(x)= \sin(x)+3, \displaystyle g(x)= \cos(x)+3, \displaystyle h(x)= \tan(x)+3

Horizontal Shift
These are functions in the form:
\displaystyle f(x)= \sin(x+r \degree), f(x)= \cos(x+r \degree), f(x)= \tan(x+r \degree)
When \displaystyle r > 0, the function shifts to the left by r degrees.
When \displaystyle r < 0, the function shifts to the right by r degrees.
Example
\displaystyle f(x)= \sin(x-30 \degree), g(x)= \cos(x-30 \degree), h(x)= \tan(x \degree).
\displaystyle r = -30, \displaystyle r < 0
The functions shifts to the right by 30 degrees. Below are the graphs.

\displaystyle f(x)= \sin(x-30 \degree), \displaystyle g(x)= \cos(x-30 \degree), \displaystyle h(x)= \tan(x-30 \degree)

Change in Amplitude
Definition: The greatest value of the the function can have is called the amplitude.
These are functions in the form:
\displaystyle f(x)= a \sin(x), f(x)= a \cos(x), where \displaystyle a \neq 1
When \displaystyle a > 1, the amplitude of the function increases.
When \displaystyle 0 < a < 1, the amplitude of the function decreases.
When \displaystyle a < 0, the function is reflected about the x-axis.
When \displaystyle -1 < a < 0 , the function is reflected about the x-axis and the amplitude decreases.
When \displaystyle a < -1, the amplitude of the function decreases.
Example
\displaystyle f(x)= 4 \sin(x), g(x)= 4 \cos(x).
\displaystyle a = 4, \displaystyle a > 1
The amplitude of the function increases.

\displaystyle f(x)= 4 \sin(x), \displaystyle g(x)= 4 \cos(x)

Change of Period
Definition: The Period goes from one peak to the next.
These are functions in the form:
\displaystyle f(x)= \sin(kx), f(x)= \cos(kx)
When \displaystyle k > 0:
\displaystyle k > 1: the period of the function decreases.
\displaystyle 0 < k < 1: the period of the function increases.

When \displaystyle k < 0:
\displaystyle -1 < k < 0: the period of the function increases. The function is reflected about the y-axis.
\displaystyle k < -1: the period of the function decreases. The function is reflected about the y-axis.
Example
\displaystyle f(x)= \sin(5x), g(x)= \cos(5x).
\displaystyle k = 5, \displaystyle k > 1
The period of the function decreases.

\displaystyle f(x)= \sin(5x), \displaystyle g(x)= \cos(5x)